Discrete Painlev\'e equations and their Lax pairs as reductions of integrable lattice equations

TL;DR

The paper introduces a method to derive Lax representations for similarity reductions of integrable lattice equations, exemplified by obtaining the q-Painlevé equation with an E_6^{(1)} affine Weyl group symmetry from the discrete Schwarzian KdV equation.

Contribution

A novel framework for deriving Lax pairs for reductions of integrable lattice equations, enabling the analysis of entire hierarchies of such reductions.

Findings

Derived Lax representations for specific reductions

Connected q-Painlevé equations to lattice equations via similarity reductions

Demonstrated the method with the discrete Schwarzian KdV and E_6^{(1)} symmetry

Abstract

We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the q-Painlev\'e equation whose group of B\"acklund transformations is an affine Weyl group of type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation.